Hans Fangohr

Summary of a recent publication based on Molecular Dynamics and Monte
Carlo simulations of particles, with contributions from University of
Massachusetts (US), The Royal Institute of Technology, Stockholm
(Sweden), The Pennsylvania State University (US), University of West
Florida (US) and the University of Southampton (UK):

Context

Lennard-Jones potential

Figure A: (Left) The Lennard Jones potential: the distance A0 is the optimum distance as it has the lowest potential energy. (Right) A purely repulsive potential: particles exposed to this will try to separate from each other as far as possible. We have used arbitrary units on both axes for these schematic plots.

The physics and emergent behaviour of interacting particles has long
standing history: classical computational problems include the
simulation of the behaviour of many atoms in liquids, solids, proteins
etc, and common simulation techniques are Molecular Dynamics and
(Metropolis) Monte Carlo methods.

For both, we need to know the pairwise interaction potential between
two particles (we ignore here systems that require 3 and more body
interactions to be considered).

For example, the well known Lennard Jones potential (shown in Figure A
(left) above) for two particles such as inert atoms, has a repulsive
term that for short distances R increases the energy (of the type
$1/R^{12}$, and an attractive term (of the type
$1/R^6$. The first term originates in strong repulsion
of electron orbitals that start to overlap, the second in weak
attraction from induced electrical polarisation. The two terms
combined result in a potential as shown in figure 1a), which has a
minimum at a distance A0 (approximately 3 for the schematic sketch):
each pair of particles has the lowest possible energy if they can be
separated by this distance A0.

Repulsive potential

Another area of complex system research that uses particle-based
simulation techniques such as Molecular Dynamics and Monte Carlo
methods is that of the the dynamics of vortex lines in (Type II)
superconductors. In these systems, the vortex lines always repel each
other, and a corresponding potential is shown in figure A (right): for
all distances, the energy decreases if we increase the distance
between the two interacting objects, separated by a distance R.

In these systems, the vortex lines cannot escape the sample, so that
they will arrange in a way to minimise they energy, which is - in the
absence of any other disordering effects and absence of geometrical
constraints of the sample - a hexagonal lattice.

Novelty in this work

Recently, the possibility of more complicated inter-vortex interactions in newly discovered systems (so-called Type 1.5 superconductors) has attracted much attention: in multi-component and multi-layer superconductors the interaction potential:

Figure C demonstrates a repulsive potential with multiple length scales: for shortest length scales, the interaction energy is high (although for the discrete function on the left it doesn't matter what the particle separation is as long as the distance is with the range for which the potential is constant), and decreases as the separation increases. The figure on the right shows a physically more realistic potential with smooth rather than discrete step changes.

Selected results

Figure D shows (left) vortex equilibrium configurations for the discrete potential and (right) for the smoothed potential, both figures revealing clusters of vortices that arrange in a hexagonal lattice. The insets show details about each cluster: for the step potential the position of the particles within the same plateau of the step function is irrelevant (left inset) where as for the smooth repulsive potential (right inset) the vortices inside the cluster try to maximise their distance. [This is taken from Figure 2 in publication.]

Figure E shows a number of configurations that are obtained for multi-length scale potentials with attractive and repulsive components. [Figure E is taken from figure 4 in publication).]